If I have four dots, arranged in two rows of two to make a square, and I draw a triangle by joining three of the dots, there are four triangles I can draw, but they are all the same shape (they are congruent). If I start with nine dots, arranged in three rows of three to make a square, how many different (non-congruent) triangles is it possible to draw? What if I start with a square formed of sixteen dots? Can you generalise for n2 dots?
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Do the triangles have to be made of as many dots as n+1 – Pi_die_die Jul 21 '18 at 10:45
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Start with a simpler problem that you cannot yet solve: How many nondegenerate triangles are there in a grid of $n^2$ points? – Christian Blatter Jul 21 '18 at 13:55
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Can the triangles be turned in space? Or only in the plane? I.e., do you distinguish them according to chirality? – joriki Jul 21 '18 at 14:38
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The triangles you count for each number of dots can not be rotations of reflection of each other. – PERCIVAL Jul 22 '18 at 13:28